Best Known (51−27, 51, s)-Nets in Base 25
(51−27, 51, 153)-Net over F25 — Constructive and digital
Digital (24, 51, 153)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (1, 14, 27)-net over F25, using
- net from sequence [i] based on digital (1, 26)-sequence over F25, using
- digital (10, 37, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- the Hermitian function field over F25 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- digital (1, 14, 27)-net over F25, using
(51−27, 51, 280)-Net over F25 — Digital
Digital (24, 51, 280)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2551, 280, F25, 2, 27) (dual of [(280, 2), 509, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2551, 313, F25, 2, 27) (dual of [(313, 2), 575, 28]-NRT-code), using
- 251 times duplication [i] based on linear OOA(2550, 313, F25, 2, 27) (dual of [(313, 2), 576, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2550, 626, F25, 27) (dual of [626, 576, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- linear OA(2550, 625, F25, 27) (dual of [625, 575, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2549, 625, F25, 26) (dual of [625, 576, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(250, 1, F25, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(250, s, F25, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- OOA 2-folding [i] based on linear OA(2550, 626, F25, 27) (dual of [626, 576, 28]-code), using
- 251 times duplication [i] based on linear OOA(2550, 313, F25, 2, 27) (dual of [(313, 2), 576, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2551, 313, F25, 2, 27) (dual of [(313, 2), 575, 28]-NRT-code), using
(51−27, 51, 56212)-Net in Base 25 — Upper bound on s
There is no (24, 51, 56213)-net in base 25, because
- 1 times m-reduction [i] would yield (24, 50, 56213)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 7888 614182 801357 191039 438732 569973 292507 660392 723025 445137 901987 878425 > 2550 [i]